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In this paper, we consider the effects of opacity regimes on the stability of self-gravitating protoplanetary discs to fragmentation into bound objects. Using a self-consistent 1D viscous disc model, we show that the ratio of local cooling to dynamical time-scales Ωtcool has a strong dependence on the local temperature. We investigate the effects of temperature-dependent cooling functions on the disc gravitational stability through controlled numerical experiments using a smoothed particle hydrodynamics code. We find that such cooling functions raise the susceptibility of discs to...

In this paper, we consider the effects of opacity regimes on the stability of self-gravitating protoplanetary discs to fragmentation into bound objects. Using a self-consistent 1D viscous disc model, we show that the ratio of local cooling to dynamical time-scales Ωtcool has a strong dependence on the local temperature. We investigate the effects of temperature-dependent cooling functions on the disc gravitational stability through controlled numerical experiments using a smoothed particle hydrodynamics code. We find that such cooling functions raise the susceptibility of discs to fragmentation through the influence of temperature perturbations – the average value of Ωtcool has to increase to prevent local variability leading to collapse. We find the effects of temperature dependence to be most significant in the ‘opacity gap’ associated with dust sublimation, where the average value of Ωtcool at fragmentation is increased by over an order of magnitude. We then use this result to predict where protoplanetary discs will fragment into bound objects, in terms of radius and the accretion rate. We find that without temperature dependence, for radii of ≲10 au, a very large accretion rate of ∼10−3 M⊙ yr−1 is required for fragmentation, but that this is reduced to 10−4 M⊙ yr−1 with temperature-dependent cooling. We also find that the stability of discs with accretion rates of ≲10−7 M⊙ yr−1 at radii of ≳50 au is enhanced by a lower background temperature if the disc becomes optically thin.